Connecting geodesics and security of configurations in compact locally symmetric spaces

A connecting theorem for geodesic flows on the 2-torus

We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle $ST^2$ of the 2-torus in regions without invariant tori.

Growth of the number of geodesics between points and insecurity for riemannian manifolds

A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics with length $\leq T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \to \infty$. We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point…

Poincar{é} series and linking of Legendrian knots

On a negatively curved surface, we show that the Poincar{é} series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincar{é} series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesi…

Connection blocking in homogeneous spaces and nilmanifolds

Let $G$ be a connected Lie group acting locally simply transitively on a manifold $M$. By connecting curves in $M$ we mean the orbits of one-parameter subgroups of $G$. To block a pair of points $m_1,m_2\in M$ is to find a finite set $B\subset M\setminus{m_1,m_2}$ such that every connecting curve joining $m_1$ and $m_2$ intersects $B$. The homogeneous space $M$ is blockable if every pair of points in $M$ can be blocked. Motivated by the geodesic security [4], we conjecture that the only blockable homogeneous spaces of finite volume are the tori. Here we establish the conjecture for nilmanifold…

On transversal connecting orbits of Lagrangian systems in non-stationary force field: Newton-Kantorovich approach

We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to the action of a non-stationary force field with potential $U(q,t) = f(t)V(q)$. It is assumed that the factor $f(t)$ tends to $\infty$ as $t\to \pm\infty$ and vanishes at a unique point $t_{0}\in \mathbb{R}$. Let $X_{+}$, $X_{-}$ denote the sets of isolated critical points of $V(x)$ at which $U(x,t)$ as a function of $x$ attains its maximum for any fixed $t> t_{0}$ and $t<t_{0}$, respectively. Under nondegeneracy conditions on points of $X_{\pm}$ we apply Newton-Kantorovich type method to study …

拉格朗日系统与测地线的分支现象 I

Bifurcations for Lagrangian systems and geodesics I

本文研究了具有不同边界条件和约束的拉格朗日系统中的分支现象，通过三种主要配置探讨了莫尔斯理论与多重解存在的关系，提出了分支条件的必要充分条件。